Qutrit Dichromatic Calculus and Its Universality

We introduce a dichromatic calculus (RG) for qutrit systems. We show that the decomposition of the qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As an application of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Since it is not easy to decompose an arbitrary d by d unitary matrix into Z and X phase gates when d > 2, the proof of the universality of qudit ZX calculus for quantum mechanics is far from trivial. We construct a counterexample to Ranchin's universality proof, and give another proof by Lie theory that the qudit ZX calculus contains all single qudit unitary transformations, which implies that qudit ZX calculus, with qutrit dichromatic calculus as a special case, is universal for quantum mechanics.Comment: In Proceedings QPL 2014, arXiv:1412.810

Generators and Relations for 3-Qubit Clifford+CS Operators

We give a presentation by generators and relations of the group of 3-qubit Clifford+CS operators. The proof roughly consists of two parts: (1) applying the Reidemeister-Schreier theorem recursively to an earlier result of ours; and (2) the simplification of thousands of relations into 17 relations. Both (1) and (2) have been formally verified in the proof assistant Agda. The Reidemeister-Schreier theorem gives a constructive method for computing a presentation of a sub-monoid given a presentation of the super-monoid. To achieve (2), we devise an almost-normal form for Clifford+CS operators. Along the way, we also identify several interesting structures within the Clifford+CS group. Specifically, we identify three different finite subgroups for whose elements we can give unique normal forms. We show that the 3-qubit Clifford+CS group, which is of course infinite, is the amalgamated product of these three finite subgroups. This result is analogous to the fact that the 1-qubit Clifford+T group is an amalgamated product of two finite subgroups.Comment: In Proceedings QPL 2023, arXiv:2308.1548

Generators and relations for 2-qubit Clifford+T operators

We give a presentation by generators and relations of the group of Clifford+T operators on two qubits. The proof relies on an application of the Reidemeister-Schreier theorem to an earlier result of Greylyn, and has been formally verified in the proof assistant Agda.Comment: 13 page

Fast and Effective Techniques for T-Count Reduction via Spider Nest Identities

In fault-tolerant quantum computing systems, realising (approximately) universal quantum computation is usually described in terms of realising Clifford+T operations, which is to say a circuit of CNOT, Hadamard, and ?/2-phase rotations, together with T operations (?/4-phase rotations). For many error correcting codes, fault-tolerant realisations of Clifford operations are significantly less resource-intensive than those of T gates, which motivates finding ways to realise the same transformation involving T-count (the number of T gates involved) which is as low as possible. Investigations into this problem [Matthew Amy et al., 2013; Gosset et al., 2014; Matthew Amy et al., 2014; Matthew Amy et al., 2018; Earl T. Campbell and Mark Howard, 2017; Matthew Amy and Michele Mosca, 2019] has led to observations that this problem is closely related to NP-hard tensor decomposition problems [Luke E. Heyfron and Earl T. Campbell, 2018] and is tantamount to the difficult problem of decoding exponentially long Reed-Muller codes [Matthew Amy and Michele Mosca, 2019]. This problem then presents itself as one for which must be content in practise with approximate optimisation, in which one develops an array of tactics to be deployed through some pragmatic strategy. In this vein, we describe techniques to reduce the T-count, based on the effective application of "spider nest identities": easily recognised products of parity-phase operations which are equivalent to the identity operation. We demonstrate the effectiveness of such techniques by obtaining improvements in the T-counts of a number of circuits, in run-times which are typically less than the time required to make a fresh cup of coffee

Finishing the euchromatic sequence of the human genome

The sequence of the human genome encodes the genetic instructions for human physiology, as well as rich information about human evolution. In 2001, the International Human Genome Sequencing Consortium reported a draft sequence of the euchromatic portion of the human genome. Since then, the international collaboration has worked to convert this draft into a genome sequence with high accuracy and nearly complete coverage. Here, we report the result of this finishing process. The current genome sequence (Build 35) contains 2.85 billion nucleotides interrupted by only 341 gaps. It covers ∼99% of the euchromatic genome and is accurate to an error rate of ∼1 event per 100,000 bases. Many of the remaining euchromatic gaps are associated with segmental duplications and will require focused work with new methods. The near-complete sequence, the first for a vertebrate, greatly improves the precision of biological analyses of the human genome including studies of gene number, birth and death. Notably, the human enome seems to encode only 20,000-25,000 protein-coding genes. The genome sequence reported here should serve as a firm foundation for biomedical research in the decades ahead